Power Series Convergence Possibilities Theorem

Theorem

For a given power series ${\sum }_{i=0}^n{C}_i(x-a)$ There are 3 possibilities:

1. Power series converges to $x=a$
2. Power series converges for all $x$
3. For some Radius of Convergence $r$, $|x-a|<r$ conv, $|x-a|>r$ div